College Algebra: Absolute Value Expressions

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Okay, so the absolute value is always positive, and let me tell you a reason why. You can actually think of the absolute value as a measuring of distance. So, in fact, if we have a number line, and I wanted to measure the distance... This is zero and this is 5. If I want to measure the distance between zero and 5 what I would do is I would take a ruler, like this, and put it down and read off what it says. In this case it would say 5. Okay, great. However, if the number were over here--this number, let's say, is -3--if I measure that distance it still would say 3, but not -3, because, of course, the ruler starts at zero. There's no negative. You can't have negative 5 inches. So, in fact, the absolute value is measuring distance, and we can actually use that in a variety of ways.
For example, if I say evaluate, or figure out, what this would equal. Let's say the absolute value of * - 3. Well, that's actually asking for the distance between * and 3. Well, what would that equal? Well, I remember that * is--you know, the thing for the circles--it's like 3.141598 and it goes on, so it's actually a little bit bigger than three. So if I take a number that's a little bit bigger than 3 and I subtract off 3, that number inside there is positive, and if that number is positive, remember what the rule for absolute value says. If you've got a positive thing in there and taking the absolute values, it's just itself--not a problem. So, in fact, this could just be written as * - 3. And that's what it would be. So a lot of times you can actually get rid of absolute values when they're there, if you just think about whether the thing is positive or negative. If the number is negative, remember the absolute value of it will be negative of that number to make it positive.
So let's do now a really, really big example. Take a look at this one. Let's take 1 and subtract off the absolute value of 4 - *. Okay, that's bad enough. But now let's divide all that by 2 times the absolute value of 3 - *. Now, that looks pretty bad in and of itself, but folks, I'm going to make life even that much worse by putting absolute values around the whole thing. Look at that. Now, that is the mother of all absolute value problems. There it is. And what I'd like to do is for us to slowly work through and simplify this and see if we can say this in a nicer way. And the secret to this kind of thing is to start on the very, very inside. You know, when you're faced with a problem in life--let me tell you something--whether it's this course or just in your life, because this course, who cares? But in your life, that's a big deal. When you're faced with a big problem you know what you should do? Try to tackle the little, teeny parts of it first. Just keep nipping at it and the next thing you know, the whole problem will fall out and be solved. You'll be rich and famous, and then send those royalty checks to me, Professor Edward Burger.
So let's use that philosophy here, and let's start with the most inside smallest components and then work our way out. So the first thing I'm going to do is take a look at these two terms here with absolute values, and let's see if we can say what those things are going to be equal. Well, here I have 4 and I'm subtracting pi. I have to determine if that number is positive or negative. Well, pi is 3.14 something, and this is 4. So, in fact, 4 minus pi is going to be a number that's actually bigger than zero. So the absolute value of it will just be itself. So I can actually remove those absolute values. And so what I see here is that this will equal--well, I have those big absolute values way out there in front--but here I'm going to have a 1 minus--and then I'm just going to have this naked, exposed to the elements. Okay? So there's the top part. And then I'm going to divide this by something on the bottom, which I'm going to write in. Are you happy now? Well, actually I hope you are not happy, because I have made a classic mistake. I have made a classic mistake. In fact, this actually makes my top ten list. In fact, do you know what classic mistake it is? I wish I did. I think it's classic mistake--let me think about this for a second. I've got to think... I think it's classic mistake number 4. Over there. You see it? I wish I could see it, but it's hidden from sight because it's on the same plane, but if I had a helper someone would go and grab it and I'd be able to see it. You can see it right there.
It's classic mistake number 4, and it's the subtracting mistake. The thing is, when you subtract something like this, you've got to subtract everybody. You see, what this is saying here is subtract off all that stuff--whatever it is, just subtract it off. And then what is this saying? What this is actually saying is, subtract off the 4 and then just do whatever you were doing before. Well, you want to subtract off all that stuff, so I need parenthesis here. In fact, I need red ones, and I need them now around this whole quantity. This is a classic mistake. It's classic mistake number 4, it's the subtracting mistake, and remember to spread the negativity. You've got to distribute that thing, not only to the four, but also to the minus pi. Really important point--great, great mistake--number 4 on my list of ten.
Okay, let's keep going. So then on the bottom here I'm going to have the 2, and let's see what this is going to equal. Well, 3 minus pi--I'm taking the absolute value of that, but notice that now I'm taking 3 and subtracting a number that's bigger than 3. So this number is negative. So to take the absolute value of it, I've got to put a negative in front of it to cancel off the negative that we have. And so what I would see here would be a negative times all that--3 minus pi.
All right, so now I've got all these negative signs running around, I've got to distribute really carefully. That negative sign has to be distributed all over here and here. And so let's try that and see what happens. What would happen is the following. Like let me just keep this up here for you so you can sort of see that and enjoy that at your leisure, as they say. Anyway, so let's see what happens here. So if I distribute--so what does this equal? Well, here I'd see a 1, and I'm going to distribute that negative sign, so that's a -4, but now a minus and a minus make a plus pi. So you see, you would have made a mistake if you would have followed my lead. I would have gotten it wrong because I would have put a minus pi and it should be a plus pi. That negative sign has to hit everything. That's on the top.
What's on the bottom? I've got a 2 times--and I've got to distribute now the negative sign here, so I see a negative 3, but now I see a plus pi. So I've got that. Now, what does that equal? Well, that equals the absolute value of--well now, 1 minus 4 is -3 plus pi, so that's pretty okay. Now, here I've got to multiply everything through by 2. Now, I could do that, and you know, sometimes in life you're tempted to do things. But you sometimes have to stop and think is that a good idea. Look, I see a -3 plus pi on the top. I've got a really nice -3 plus pi on the bottom. This looks like it's sort of set for a nice cancellation. So, in fact, I'm going to suppress the desire to expand the bottom. And you know what? A lot of times we should try to keep the bottom as factored as we can. And so if life hands us a factored denominator, let's not go and mess that up by actually expanding it. Let's keep it factored. And you'll see the value of that right now. This is great. This is full of life lessons.
Because now notice the whole top and this little factor on the bottom can be cancelled. So I can cancel this whole top with this. And what does that leave me with? Well, you may think it leaves me with zero on top. But remember, always in life you're followed by an invisible multiple of 1. So, in fact, really here, there's a 1 on top. So we have a 1 on top and we have a 2 on the bottom, and don't forget those big absolute values.
Remember how big those absolute values were at the start of this problem? They were this big. And look how we tamed them into size. Now they're only this big. See? Not a problem. And I happen to know what the sequel is--I worked this out in advance. The absolute value of a half turns out to be half.
So you can see how working through a really, really hard, complicated problem like that, if we take it slowly, work from the inside out, we can actually achieve the solution.
What I want to take a look at now is a problem, or the kind of issue where, in fact, you're given some variables. Now, in that previous example, of course, everything was known. We just had all the numbers. The numbers were there; we just plowed through. But what if you don't know all the numbers? Did you ever think of that? All right. Well, let's think about it now.
For example, suppose I don't tell you exactly what a number is, I just tell you that the number, let's call it t, is some number that's less than 3. And now what I want you to do is to figure out what the absolute value of t minus 3 is. Well, how would you do that problem? Well, what we have to do is use this information to figure out if this thing is positive or negative. Now, let's think about this. If t is a number less than 3, and I take t and subtract 3, what kind of number is that? Well, I can just think about that a little bit and reason that that must, in fact, be negative. But I can actually do it algebraically. Let me do it for you really fast algebraically. What I would do is the following:
I would start off with what I know, so what I know is that t is less than 3. I'm interested in t minus 3, so you know what I'll do? I'll use those properties of inequalities that we know, and I'm going to subtract 3 from both sides. Let's see what happens. If I subtract 3 from both sides, what I see is t minus 3 would still be less than--and now I see 3 minus 3, zero. And so now my intuition originally was right. This number is a negative number. So to take absolute values of it, what do I do? I smash in another negative sign in front of it, and that's going to make the whole thing positive. So this answer would be negative of t minus 3. Notice the parenthesis here, folks. This is really great. I am not making classic mistake number 4. And so if I distribute I would see -t and then a minus, a minus is a plus 3. This is where the mistake usually is made--someone puts a -3 there. I hope you make that mistake right now. Get it out of your system and always distribute again. So the answer here will be -t + 3, or you could say 3 - T.
Let's try one last one really fast to drill this home. Let's suppose that now we're told that some number, x, is smaller than -2. Let's see if we can figure out what the absolute value of 6 - 5x would be. Now, this is a little bit harder than the last one. The last one I was able to reason and figure out whether that's positive or negative. Now this is a little bit harder, because I just can't reason that much. I'll have to start with what I know and try to massage that into looking like this and seeing if it's positive or negative. So let's do it together.
So I know that x is less than or smaller than -2. So what do I want? I want to now look at this thing. So the first thing I want to do is get a -5x in the picture. So I'm going to multiply this through by -5. Now, remember, of my classic mistake number 7, multiplying an inequality mistake. When I multiply and inequality by a negative number, it's going to switch that sign. So what I see here is -5x, and now that flips and becomes now a greater than, and a -5 times -2 is positive 10. So now I've got that going on, which is good, but I actually want a 6 there. So I'll just add 6 to both sides. Remember, adding 6 doesn't hurt anything. The sign stays the same. And I see that 6 - 5x is going to be bigger than 16. Well, all I care about is whether it's positive or negative. Well, now I see it's extremely positive; it's even bigger than 16. So what's the absolute value of it? Since it's positive, it's just itself. And so the absolute value of 6 - 5x is just itself. 6 - 5x in the case when x is smaller than -2.
There you have more absolute values than you ever thought humanly possible. Enjoy.
Prerequisites
Absolute Values
Evaluating Absolute Value Expressions Page [1 of 3]